Comment on “A constructive proof on the existence of globally exponentially attractive set and positive invariant set of general Lorenz family”, P. Yu, X.X. Liao, S.L. Xie, Y.L. Fu [Commun Nonlinear Sci Numer Simulat 14 (2009) 2886–2896]

2014 ◽  
Vol 19 (3) ◽  
pp. 758-761 ◽  
Author(s):  
Antonio Algaba ◽  
Fernando Fernández-Sánchez ◽  
Manuel Merino ◽  
Alejandro J. Rodríguez-Luis
Keyword(s):  
Invariant Set ◽  
Constructive Proof ◽  
Positive Invariant Set

scholarly journals Low Model Analysis and Synchronous Simulation of the Wave Mechanics

2016 ◽  
Vol 2016 ◽  
pp. 1-13
Author(s):  
Wenyuan Duan ◽  
Heyuan Wang ◽  
Meng Kan
Keyword(s):  
Numerical Simulation ◽  
Lyapunov Function ◽  
Chaotic System ◽  
Invariant Set ◽  
Positive Definite ◽  
Chaotic Attractors ◽  
Wave Mechanics ◽  
Wave Dynamics ◽  
Simulation Results ◽  
Positive Invariant Set

The dynamic behavior of a chaotic system in the internal wave dynamics and the problem of the tracing and synchronization are investigated, and the numerical simulation is carried out in this paper. The globally exponentially attractive set and positive invariant set of the chaotic system are studied via constructing the positive definite and radial unbounded Lyapunov function. There are no equilibrium positions, periodic solutions, quasi-period motions, wandering recovering motions, and other chaotic attractors of the system out of the globally exponentially attractive set. Strange attractors can only locate in the globally exponentially attractive set. A feedback controller is designed for the chaotic system to realize the control of the unstable point. The second method of Lyapunov is used to discuss theoretically the rationality of the design of the controller. The driving-response synchronization method is used to realize the globally exponential synchronization. The numerical simulation is carried out by MATLAB software, and the simulation results show that the method is effective.

GLOBALLY EXPONENTIALLY ATTRACTIVE SETS AND POSITIVE INVARIANT SETS OF THREE-DIMENSIONAL AUTONOMOUS SYSTEMS WITH ONLY CROSS-PRODUCT NONLINEARITIES

2013 ◽  
Vol 23 (01) ◽  
pp. 1350007 ◽  
Author(s):  
XINQUAN ZHAO ◽  
FENG JIANG ◽  
JUNHAO HU
Keyword(s):  
Chaotic Systems ◽  
Autonomous Systems ◽  
Three Dimensional ◽  
Sufficient Conditions ◽  
Cross Product ◽  
Invariant Set ◽  
Invariant Sets ◽  
Special Cases ◽  
Attractive Sets ◽  
Positive Invariant Set

In this paper, the existence of globally exponentially attractive sets and positive invariant sets of three-dimensional autonomous systems with only cross-product nonlinearities are considered. Sufficient conditions, which guarantee the existence of globally exponentially attractive set and positive invariant set of the system, are obtained. The results of this paper comprise some existing relative results as in special cases. The approach presented in this paper can be applied to study other chaotic systems.

GLOBALLY ATTRACTIVE AND POSITIVE INVARIANT SET OF THE LORENZ SYSTEM

2006 ◽  
Vol 16 (03) ◽  
pp. 757-764 ◽  
Author(s):  
PEI YU ◽  
XIAOXIN LIAO
Keyword(s):  
Lyapunov Function ◽  
Chaos Synchronization ◽  
Chaos Control ◽  
Lorenz System ◽  
Equilibrium Points ◽  
Invariant Set ◽  
The Lorenz System ◽  
Globally Attractive ◽  
Generalized Lyapunov Function ◽  
Positive Invariant Set

In this paper, based on a generalized Lyapunov function, a simple proof is given to improve the estimation of globally attractive and positive invariant set of the Lorenz system. In particular, a new estimation is derived for the variable x. On the globally attractive set, the Lorenz system satisfies Lipschitz condition, which is very useful in the study of chaos control and chaos synchronization. Applications are presented for globally, exponentially tracking periodic solutions, stabilizing equilibrium points and synchronizing two Lorenz systems.

New Estimations of Bounds for the Chu Chaotic System

2011 ◽  
Vol 128-129 ◽  
pp. 196-199
Author(s):  
Ji Gui Jian ◽  
Wei Wei Wang
Keyword(s):  
Chaotic System ◽  
Function Theory ◽  
Continuous Extension ◽  
Invariant Set ◽  
Extension Principle ◽  
Cylindrical Domain ◽  
Inequality Techniques ◽  
New Ellipsoid ◽  
Generalized Lyapunov Function ◽  
Positive Invariant Set

This paper is concerned with positive invariant set for the Chu chaotic system using a technique combing the generalized Lyapunov function theory and maximum principle. For this chaotic system, some new ellipsoid estimations and a cylindrical domain estimation of the globally exponentially attractive set for all the positive parameters values of the system are derived without existence assumptions via employing inequality techniques and the continuous extension principle.

NEW ESTIMATIONS FOR GLOBALLY ATTRACTIVE AND POSITIVE INVARIANT SET OF THE FAMILY OF THE LORENZ SYSTEMS

2006 ◽  
Vol 16 (11) ◽  
pp. 3383-3390 ◽  
Author(s):  
PEI YU ◽  
XIAOXIN LIAO
Keyword(s):  
Lyapunov Functions ◽  
Chaotic Systems ◽  
Invariant Set ◽  
The Other ◽  
Qualitative Behavior ◽  
The Family ◽  
Infinite Intervals ◽  
Globally Attractive ◽  
Generalized Lyapunov Functions ◽  
Positive Invariant Set

In this paper, we employ generalized Lyapunov functions to derive new estimations of the ultimate boundary for the trajectories of two types of Lorenz systems, one with parameters in finite intervals and the other in infinite intervals. The new estimations improve the results reported so far in the literature. In particular, for the singular cases: b → 1+ and a → 0+, we have obtained the estimations independent of a. Moreover, our method using elementary algebra greatly simplifies the proofs in the literature. This is an interesting attempt in obtaining information of the attractors which is difficult when merely based on differential equations. It indicates that Lyapunov function is still a powerful tool in the study of qualitative behavior of chaotic systems.

LOCALIZATION OF ATTRACTORS IN MULTIDIMENSIONAL CHUA'S CIRCUIT EQUATIONS

1993 ◽  
Vol 03 (02) ◽  
pp. 497-506 ◽  
Author(s):  
G. A. LEONOV ◽  
V. B. SMIRNOVA
Keyword(s):  
Parameter Space ◽  
Basin Of Attraction ◽  
Invariant Set ◽  
Chua’S Circuit ◽  
Chua's Circuit ◽  
Positive Invariant Set

The well-known double-scroll Chua's attractor occurs in the parameter space where all 3 linearized regions of the dimensionless Chua's equation are unstable. To investigate the surprising existence of a basin of attraction, this paper proves a theorem which can be used to derive a bounded positive invariant set in Chua's circuit.

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